Crowdsourcing and All-Pay Auctions

Milan Vojnovic

Microsoft Research

Lecture series – Contemporary Economic Issues – University of East Anglia, Norwich, UK, November 10, 2014

This Talk

• An overview of results of a model of competition-based crowdsourcing services based on all-pay auctions

• Based on lecture notes Contest Theory, V., course, Mathematical TriposPart III, University of Cambridge - forthcoming book

2

Competition-based Crowdsourcing: An Example

3

CrowdFlower

Statistics

• TopCoder data covering a ten-year period from early 2003 until early 2013

• Taskcn data covering approximately a seven-year period from mid 2006 until early 20134

Example Prizes: TopCoder

5

Example Participation: Tackcn

• A month in year 20106

players

contests

Game: Standard All-Pay Contest• 𝑛 players, 𝑣1, 𝑣2, … , 𝑣𝑛 valuations, linear production costs

• Quasi-linear payoff functions: 𝑠𝑖 𝒃 = 𝑣𝑖𝑥𝑖 𝒃 − 𝑏𝑖

• Simultaneous effort investments: 𝒃 = 𝑏1, 𝑏2, … , 𝑏𝑛 ,𝑏𝑖 = effort investment of player 𝑖

• Winning probability of player 𝑖: 𝑥𝑖(𝒃)highest-effort player wins with uniform random tie break

1 2 𝑛

⋮7

Strategic Equilibria

• A pure-strategy Nash equilibrium does not exist

• In general there exists a continuum of mixed-strategy Nash equilibrium

Moulin (1986), Dasgupta (1986), Hillman and Samet (1987), Hillman and Riley (1989), Ellingsen(1991), Baye et al (1993), Baye et al (1996)

• There exists a unique symmetric Bayes-Nash equilibrium

8

Symmetric Bayes-Nash Equilibrium

• Valuations 𝑣1, 𝑣2, … , 𝑣𝑛 are assumed to be private information of players, and independent samples from a prior distribution 𝐹 on [0,1]

• A strategy 𝛽: 0,1 → 0,1 is a symmetric Bayes-Nash equilibrium if it is a best response for every player 𝑖 conditional on that all other players play strategy 𝛽, i.e.

𝐄 𝑠𝑖 𝒃 𝑣𝑖 = 𝑣, 𝑏𝑖 = 𝛽 𝑣 ≥ 𝐄 𝑠𝑖 𝒃 𝑣𝑖 = 𝑣, 𝑏𝑖′ , for every 𝑣 and 𝑏𝑖′

𝛽 𝑣 = 0

𝑣

𝑥𝑑𝐹𝑛−1(𝑥)

9

Quantities of Interest

• Expected total effort: 𝑅 = 𝐄 𝑖=1𝑛 𝑏𝑖

• Expected maximum individual effort: 𝑅 = 𝐄[max{𝑏1, 𝑏2, … , 𝑏𝑛}]

• Social efficiency: 𝐄 𝑖=1

𝑛 𝑣𝑖𝑥𝑖 𝒃

𝐄[𝑣(𝑛,1)]

Order statistics: 𝑣(𝑛,1) ≥ 𝑣 𝑛,2 ≥ ⋯ ≥ 𝑣 𝑛,𝑛 (valuations sorted in decreasing order)

10

Quantities of Interest (cont’d)

• In the symmetric Bayes-Nash equilibrium:

𝑅 = 𝐄 𝑣 𝑛,2

𝑅1 = 𝐄 𝑣 𝑛−1,1 −𝑛 − 1

2𝑛 − 1𝐄[𝑣(2𝑛−1,1)]

11

Total vs. Max Individual Effort

• In any symmetric Bayes-Nash equilibrium, the expected maximum individual effort is at least half of the expected total effort

𝑅1 ≥ 𝑅/2

Chawla, Hartline, Sivan (2012)

12

Contests that Award Several Prizes: Examples

Kaggle TopCoder

13

Rank Order Allocation of Prizes• Suppose that the prizes of values 𝑤1 ≥ 𝑤2 ≥ ⋯ ≥ 𝑤𝑛 ≥ 0 are allocated to

players in decreasing order of individual efforts

• There exists a symmetric Bayes-Nash equilibrium given by

𝛽 𝑣 = 𝑗=1𝑛−1(𝑤𝑗 − 𝑤𝑗+1) 0

𝑣𝑥𝑑𝐹𝑛−1,𝑗(𝑥)

• 𝐹𝑚,𝑗 = distribution of the value of 𝑗-th largest valuation from 𝑚 independent samples from distribution 𝐹

• Special case: single unit-valued prize 1 = 𝑤1 > 𝑤2 = ⋯ = 𝑤𝑛 = 0 boils down to symmetric Bayes-Nash equilibrium in slide 9

14V. – Contest Theory (2014)

Rank Order Allocation of Prizes (cont’d)

• Expected total effort:

𝑅 = 𝑗=1𝑛−1 𝑤𝑗 − 𝑤𝑗+1 𝐄[𝑣(𝑛,𝑗+1)]

• Expected maximum individual effort:

𝑅1 = 𝑗=1𝑛−1 𝑤𝑗 −𝑤𝑗+1 𝐄 𝑣 𝑛−1,𝑗 −

𝑛−1 ! 2𝑛−1−𝑗 !

𝑛−1−𝑗 ! 2𝑛−1 !𝐄 𝑣 2𝑛−1,𝑗

15V. – Contest Theory (2014)

The Limit of Many Players

• Suppose that for a fixed integer 𝑘: 𝑤1 ≥ 𝑤2 ≥ ⋯ ≥ 𝑤𝑘 > 𝑤𝑘+1 = ⋯ = 𝑤𝑛 = 0

• Expected individual efforts:

lim𝑛→∞

𝐄 𝑏 𝑛,𝑖 = 𝑗=1𝑘 1

2𝑗+𝑖−1

𝑗 + 𝑖 − 2𝑗 − 1

𝑤𝑗

• Expected total effort:lim𝑛→∞

𝑹 = 𝑗=1𝑘 𝑤𝑗

• In particular, for the case of a single unit-valued prize (𝑘 = 1):

lim𝑛→∞

𝐄 𝑏 𝑛,𝑖 =1

2𝑖

16Archak and Sudarajan (2009)

When is it Optimal to Award only the First Prize?

• In symmetric Bayes-Nash equilibrium both expected total effort and expected maximum individual effort achieve largest values by allocating the entire prize budget to the first prize.

• Holds more generally for increasing concave production cost functions

Moldovanu and Sela (2001) – total effort

Chawla, Hartline, Sivan (2012) – maximum individual effort 17

Importance of Symmetric Prior Beliefs

• If the prior beliefs are asymmetric then it can be beneficial to offer more than one prize with respect to the expected total effort

• Example: two prizes and three players

Values of prizes w1, w2 = 𝑤, 1 − 𝑤Valuations of players 𝑣 = 𝑣1 > 𝑣2 = 𝑣3 = 1

Mixed-strategy Nash equilibrium in the limit of large 𝑣:

18V. - Contest Theory (2014)

Optimal Auction

• Virtual valuation function: 𝜓 𝑣 = 𝑣 −1−𝐹(𝑣)

𝑓(𝑣)

• 𝐹 said to be regular if it has increasing virtual valuation function

• Optimal auction w.r.t. profit to the auctioneer: (𝑥, 𝑝)

Allocation 𝑥 maximizes

𝐄 𝑖=1𝑛 𝜓 𝑣𝑖 𝑥𝑖 𝒗

payments 𝑝𝑖 𝒗 = 𝑣𝑖𝑥𝑖 𝒗 − 0𝑣𝑖 𝑥𝑖 𝒗−𝑖 , 𝑣 𝑑𝑣

Myerson (1981)19

Optimal All-Pay Contest w.r.t. Total Effort

• Suppose 𝐹 is regular. Optimal all-pay contest allocates the prize to a player who invests the largest effort subject to a minimum required effort of value 𝜓−1 0 .

• Example: uniform distribution: minimum required effort = 1/2

• If 𝐹 is not regular, then an “ironing” procedure can be used

20

Optimal All-Pay Contest w.r.t. Max Individual Effort

• Virtual valuation: 𝜓𝑛 𝑣 = 𝑣𝐹 𝑣 𝑛−1 −1−𝐹 𝑣 𝑛

𝑛𝑓(𝑣)

• 𝐹 is said to be regular if 𝜓𝑛(𝑣) is an increasing function

• Suppose 𝐹 is regular. Optimal all-pay contest allocates the prize to a player who invests the largest effort subject to a minimum required effort of value 𝜓𝑛

−1 0 𝐹𝑛−1 𝜓𝑛−1 0

• Example: uniform distribution: minimum required effort = 1/(𝑛 + 1)

Chawla, Hartline, Sivan (2012)21

Simultaneous All-Pay Contests

players

contests

22

Game: Simultaneous All-Pay Contests

• Suppose players have symmetric valuations (for now)

• Each player participates in one contest

• Contests are simultaneously selected by the players

• Strategy of player 𝑖: 𝑎𝑖 , 𝑏𝑖

𝑎𝑖 = contest selected by player 𝑖𝑏𝑖 = amount of effort invested by player 𝑖

23

Mixed-Strategy Nash Equilibrium

• There exists a symmetric mixed-strategy Nash equilibrium in which each player selects the contest to participate according to distribution 𝒑 given by

𝑝𝑗 = 1 − 1 −1

𝑚Φ 𝑚

1

𝑤𝑗

1/(𝑛−1)

𝑗 = 1,2,… , 𝑚

0 o.w.

• Φ𝑗 =1

1

𝑗 𝑙=1𝑗 1

𝑤𝑙

1/(𝑛−1) , 𝑗 = 1,2,… ,𝑚

• 𝑚 = max 𝑗|𝑤𝑗1/(𝑛−1)

> 1 −1

𝑗Φ𝑗

24

V. – Contest Theory (2014)

Quantities of Interest

• Expected total effort is at least ¼ of the benchmark value

𝑅∗ = 𝑗=1𝑘 𝑤𝑗 where 𝑘 = min{𝑚, 𝑛}

• Expected social welfare is at least 1 − 1/𝑒 of the optimum social welfare

𝑊∗ = 𝑗=1𝑘 𝑤𝑗

25

V. – Contest Theory (2014)

Bayes Nash Equilibrium• Contests partitioned into classes based on values of prizes: contests of

class 1 offer the highest prize value, contests of class 2 offer the second highest prize value, …

• Suppose valuations are private information and are independent samples from a prior distribution 𝐹

• In symmetric Bayes Nash equilibrium, players are partitioned into classes such that a player of class 𝑙 selects a contest of class 𝑗 with probability

𝛼𝑗𝑙 =

Φ𝑙

𝑀𝑙

1

𝑤𝑗

1/(𝑛−1)

1 ≤ 𝑗 ≤ 𝑙

1 o.w.

DiPalantino and V. (2009)

26𝑀𝑙 = number of contests of class 1 through 𝑙

Example: Two Contests

• 𝑎 =𝑤2

𝑤1

1/(𝑛−1)

Class 1 equilibrium strategy Class 2 equilibrium strategy

27V. – Contest Theory (2014)

Participation vs. Prize Value

• Taskcn 2009 – logo design tasks

any rate once a month every fourth day every second day

28model

DiPalantino and V. (2009)

Conclusion

• A model is presented that is a game of all-pay contests

• An overview of known equilibrium characterization results is presented for the case of the game with incomplete information, for both single contest and a system of simultaneous contests

• The model provides several insights into the properties of equilibrium outcomes and suggests several hypotheses to test in practice

29

Not in this Slide Deck

• Characterization of mixed-strategy Nash equilibria for standard all-pay contests

• Consideration of non-linear production costs, e.g. players endowed with effort budgets (Colonel Blotto games)

• Other prize allocation mechanisms – e.g. smooth allocation of prizes according to the ratio-form contest success function (Tullock) and the special case of proportional allocation

• Productive efforts – sharing of a utility of production that is a function of the invested efforts

• Sequential effort investments

30

References• Myerson, Optimal Auction Design, Mathematics of Operations Research, 1981

• Moulin, Game Theory for the Social Sciences, 1986

• Dasgupta, The Theory of Technological Competition, 1986

• Hillman and Riley, Politically Contestable Rents and Transfers, Economics and Politics, 1989

• Hillman and Samet, Dissipation of Contestable Rents by Small Number of Contestants, Public Choice, 1987

• Glazer and Ma, Optimal Contests, Economic Inquiry, 1988

• Ellingsen, Strategic Buyers and the Social Cost of Monopoly, American Economic Review, 1991

• Baye, Kovenock, de Vries, The All-Pay Auction with Complete Information, Economic Theory 1996

31

References (cont’d)

• Moldovanu and Sela, The Optimal Allocation of Prizes in Contests, American Economic Review, 2001

• DiPalantino and V., Crowdsourcing and All-Pay Auctions, ACM EC 2009

• Archak and Sundarajan, Optimal Design of Crowdsourcing Contests, Int’l Conf. on Information Systems, 2009

• Archak, Money, Glory and Cheap Talk: Analyzing Strategic Behavior of Contestants in Simultaneous Crowsourcing Contests on TopCoder.com, WWW 2010

• Chawla, Hartline, Sivan, Optimal Crowdsourcing Contests, SODA 2012

• Chawla and Hartline, Auctions with Unique Equilibrium, ACM EC 2013

• V., Contest Theory, lecture notes, University of Cambridge, 2014

32